A fair coin is tossed repeatedly until either heads comes up three times in a row or tails comes up three times in a row. What is the probability that the coin will be tossed more than $10$ times? Express your answer as a common fraction.
I think we could do this by finding the complement with casework on $3,4,5,\dots,10$ tries until we flip heads/tails three times in a row, but this sounds tedious. Is there a better way?
You can use recurrence relations to count the number of $10$ toss strings without three in a row. Let $A(n)$ be the number of strings of length $n$ that end with a single $H$ or $T$ and $B(n)$ count the number of strings of length $n$ that end with $HH$ or $TT$. Then $$A(1)=2, B(1)=0 \\ A(n+1)=A(n)+B(n) \\ B(n+1)=A(n)\\A(n+1)=A(n)+A(n-1)$$ and they are just twice the Fibonacci numbers. This gives $A(10)=110, B(10)=68$ and there are $178$ total $10$ flip strings without three in a row, a chance of $\frac {178}{1024}\approx 0.174$
This is not an answer to the question.
In fact, this long-winded response does not even approach becoming an answer.
However, the (advanced) topic in this response is very relevant to the posted question.
There is a controversial issue inherent in the posted question: should Inclusion-Exclusion be used to attack this type of question? See this article for an introduction to Inclusion-Exclusion. Then, see this answer for an explanation of and justification for the Inclusion-Exclusion formula.
In my opinion, there is no easy answer to that question. Following the syntax in the 2nd Inclusion-Exclusion link above, you will find that considerations of symmetry break down when computing $~T_r ~: ~r \geq 2.$
So, under normal circumstances, this would indicate that Inclusion-Exclusion should not be used, because the computation of each of $~T_2, T_3, \cdots ~$ would have to be done manually.
However, the situation is not that simple.
Often, in these types of situations (and this particular problem is not an exception), Stars and Bars theory may be used internally to facilitate analytical (rather than manual) computation of each of $~T_2, T_3, \cdots.$ The downside is that this approach requires a deep intuitive understanding of the overall method.
For Stars and Bars theory, see this article and this article.
For corresponding illustrative problems, see this answer, this answer, or Method 2 only of this answer.
